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The coordinates of A, B, and C in the diagram are A(p, 4), B(6, 1), and C(9, q).

Which equation correctly relates p and q?

Hint: Since [tex]\(\overleftrightarrow{A B}\)[/tex] is perpendicular to [tex]\(\overleftrightarrow{B C}\)[/tex], the slope of [tex]\(\overleftrightarrow{A B}\)[/tex] times the slope of [tex]\(\overleftrightarrow{B C}\)[/tex] = -1.

A. [tex]\(q - p = 7\)[/tex]
B. [tex]\(-q - p = 7\)[/tex]
C. [tex]\(p - q = 7\)[/tex]
D. [tex]\(p + q = 7\)[/tex]

Sagot :

GinnyAnswer
To determine the relationship between [tex]\( p \)[/tex] and [tex]\( q \)[/tex] given that line segment [tex]\( \overleftrightarrow{AB} \)[/tex] is perpendicular to line segment [tex]\( \overleftrightarrow{BC} \)[/tex], we need to calculate the slopes of these line segments and use the condition that the product of their slopes equals [tex]\(-1\)[/tex].

Here are the steps:

### Step 1: Find the slope of [tex]\( \overleftrightarrow{AB} \)[/tex]
The slope of a line passing through two points [tex]\( \left(x_1,y_1\right) \)[/tex] and [tex]\( \left(x_2,y_2\right) \)[/tex] is given by:

[tex]\[ \text{slope of } \overleftrightarrow{AB} = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]

For points [tex]\( A(p, 4) \)[/tex] and [tex]\( B(6, 1) \)[/tex]:

[tex]\[ \text{slope of } \overleftrightarrow{AB} = \frac{1 - 4}{6 - p} = \frac{-3}{6 - p} \][/tex]

### Step 2: Find the slope of [tex]\( \overleftrightarrow{BC} \)[/tex]
Similarly, using points [tex]\( B(6, 1) \)[/tex] and [tex]\( C(9, q) \)[/tex]:

[tex]\[ \text{slope of } \overleftrightarrow{BC} = \frac{q - 1}{9 - 6} = \frac{q - 1}{3} \][/tex]

### Step 3: Use the perpendicular slopes condition
Since [tex]\( \overleftrightarrow{AB} \)[/tex] is perpendicular to [tex]\( \overleftrightarrow{BC} \)[/tex], their slopes multiply to give [tex]\(-1\)[/tex]:

[tex]\[ \left( \frac{-3}{6 - p} \right) \times \left( \frac{q - 1}{3} \right) = -1 \][/tex]

### Step 4: Simplify the equation
Simplify the product of the slopes and solve for [tex]\( q \)[/tex]:

[tex]\[ \frac{-3(q - 1)}{3(6 - p)} = -1 \][/tex]

[tex]\[ \frac{-3(q - 1)}{3(6 - p)} = \frac{-3(q - 1)}{18 - 3p} = -1 \][/tex]

Now multiply both sides by [tex]\( 18 - 3p \)[/tex] to clear the fraction:

[tex]\[ -3(q - 1) = -1 \times (18 - 3p) \][/tex]

[tex]\[ -3(q - 1) = -(18 - 3p) \][/tex]

### Step 5: Expand and simplify
Distribute the -1 on the right side:

[tex]\[ -3q + 3 = -18 + 3p \][/tex]

Combine like terms:

[tex]\[ 3p - 3q = 18 + 3 \][/tex]

[tex]\[ 3p - 3q = 21 \][/tex]

Divide both sides by 3:

[tex]\[ p - q = 7 \][/tex]

### Conclusion
The correct equation that relates [tex]\( p \)[/tex] and [tex]\( q \)[/tex] is:

[tex]\[ p - q = 7 \][/tex]

Therefore, the correct option is:

C. [tex]\( p - q = 7 \)[/tex]
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